If the minimum and the maximum values of the function f:[ π/4,π/2 ] → R, defined by
f( θ ) = | - sin ²θ, - 1 - sin ²θ, 1; - cos ²θ, - 1 - cos ²θ, 1; 12, 10, - 2 | are m and M respectively, then the ordered pair (m,M) is equal to :

Solution

Maths sir · 4 steps

Subtract column 1 from column 2 and the whole determinant collapses to 4 cos 2θ.

1. Clean column 2. Do C₂ → C₂ - C₁. The sin² θ and cos² θ terms cancel, so column 2 becomes -1, -1, -2.
2. Clear row 1. Now do R₁ → R₁ - R₂. Row 1 becomes (cos² θ - sin² θ, 0, 0) = (cos 2θ, 0, 0), so expand along it.

   f(θ) = cos 2θ |-1, 1; -2, -2| = cos 2θ (2 + 2) = 4 cos 2θ

3. Use the domain. For θ ∈ [π/4, π/2], 2θ ∈ [π/2, π], so cos 2θ runs from 0 down to -1. Then 4 cos 2θ runs from 0 down to -4.
4. So. m = -4 (at θ = π/2) and M = 0 (at θ = π/4), so (m, M) = (-4, 0), option (b).

The idea to keep. When two rows or columns differ only in a few terms, subtract them first. Zeros make the expansion easy, and the range then comes from the given domain, not the full range.

Answer from the NTA final answer key. Solution by PrepWiser.